Illustrative Math Alignment: Grade 7 Unit 5

Topic

Rationals and Radicals

Description

Rationalizing an Expression involves removing radicals in the denominator by multiplying by a conjugate term. For example, 1 / (√2 + 1) is rationalized by multiplying by (√2 - 1). Focuses on simplifying expressions to remove radicals, aiding in creating standard forms for fractions.

Topic

Rationals and Radicals

Description

Rationalizing an Expression involves removing radicals in the denominator by multiplying by a conjugate term. For example, 1 / (√2 + 1) is rationalized by multiplying by (√2 - 1). Focuses on simplifying expressions to remove radicals, aiding in creating standard forms for fractions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

The conjugate of a radical is a factor that eliminates radicals in an expression. For example, the conjugate of √2 is √2, and the conjugate of (1 + √2) is (1 - √2), leading to simplified results like 1 - 2 = -1. Conjugates are used to rationalize denominators and simplify expressions involving radicals.

Topic

Rationals and Radicals

Description

The conjugate of a radical is a factor that eliminates radicals in an expression. For example, the conjugate of √2 is √2, and the conjugate of (1 + √2) is (1 - √2), leading to simplified results like 1 - 2 = -1. Conjugates are used to rationalize denominators and simplify expressions involving radicals.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rationalizing the Denominator removes radicals from the denominator by multiplying by its conjugate. For example, 1 / √2 becomes √2 / 2 after rationalization. Illustrates how rationalization is used to make fractions easier to interpret and solve.

Topic

Rationals and Radicals

Description

Rationalizing the Denominator removes radicals from the denominator by multiplying by its conjugate. For example, 1 / √2 becomes √2 / 2 after rationalization. Illustrates how rationalization is used to make fractions easier to interpret and solve.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

A Square Root is a number that, when multiplied by itself, equals a given number. Square roots can be written using the radical symbol or as a rational exponent. For example, √25 = 5 and 36(1/2) = 6. This definition includes properties such as √a * √a = a. This term connects to the topic by introducing square roots, which are frequently encountered in operations involving radicals and rational expressions.

Topic

Rationals and Radicals

Description

A Square Root is a number that, when multiplied by itself, equals a given number. Square roots can be written using the radical symbol or as a rational exponent. For example, √25 = 5 and 36^(1/2) = 6. This definition includes properties such as √a * √a = a. This term connects to the topic by introducing square roots, which are frequently encountered in operations involving radicals and rational expressions.

Topic

Rationals and Radicals

Description

The Square Root Property states that if a2 = b, then a = ±√b. Examples include 52 = 25, so √25 = ±5, and 62 = 36, so √36 = ±6. This property provides the basis for solving quadratic equations and simplifying radical expressions, tying back to the earlier concepts of radicals.

Topic

Rationals and Radicals

Description

The Square Root Property states that if a2 = b, then a = ±√b. Examples include 52 = 25, so √25 = ±5, and 62 = 36, so √36 = ±6. This property provides the basis for solving quadratic equations and simplifying radical expressions, tying back to the earlier concepts of radicals.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Subtraction of Rational Expressions involves subtracting one rational expression from another, resulting in another rational expression. The example simplifies 1/(x + 1) - 1/(x - 1) into -2/(x2 - 1) by finding a common denominator and combining the terms. This term supports the topic by illustrating how operations on rational expressions, like subtraction, can be performed systematically.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

A Vertical Asymptote is defined by the restricted values of x where the function approaches but does not touch. For example, in y = 1/(x - 2), the function is not defined at x = 2, resulting in a vertical asymptote at that point. This term relates to rational expressions by identifying points of discontinuity, which are critical in understanding the behavior of rational functions.

Topic

Rationals and Radicals

Description

A Vertical Asymptote is defined by the restricted values of x where the function approaches but does not touch. For example, in y = 1/(x - 2), the function is not defined at x = 2, resulting in a vertical asymptote at that point. This term relates to rational expressions by identifying points of discontinuity, which are critical in understanding the behavior of rational functions.

Topic

Rationals and Radicals

Description

The Zero of a Rational Function is the value of the input variable for which the rational function equals zero. For example, in f(x) = (x - 1)/(x + 1), the zero occurs at x = 1. This term highlights the connection between the numerator of a rational function and its roots, which are essential for solving equations involving rational expressions.

Topic

Rationals and Radicals

Description

The Zero of a Rational Function is the value of the input variable for which the rational function equals zero. For example, in f(x) = (x - 1)/(x + 1), the zero occurs at x = 1. This term highlights the connection between the numerator of a rational function and its roots, which are essential for solving equations involving rational expressions.

Topic

Rationals and Radicals

Description

The cube root of a number, when multiplied by itself three times, equals the given number. Examples include ∛27 = 3, ∛125 = 5, and ∛8 = 2. Cube roots are essential for solving equations involving cubic powers and for understanding radical expressions.

Understanding this video is essential for grasping the fundamentals of Rationals and Radicals. The mathematical concepts explored here form the basis for further exploration and application in related fields.

Topic

Rationals and Radicals

Description

The cube root of a number, when multiplied by itself three times, equals the given number. Examples include ∛27 = 3, ∛125 = 5, and ∛8 = 2. Cube roots are essential for solving equations involving cubic powers and for understanding radical expressions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Division of rational expressions involves dividing one rational expression by another, resulting in a new rational expression. Examples include 1 / (2x2) ÷ 1 / (3x) = 3 / 2x and 2 / (x + 1) ÷ (3 / (x - 1)) = 2(x - 1) / 3(x + 1). This operation is crucial for manipulating and simplifying rational expressions in algebra.

Topic

Rationals and Radicals

Description

Division of rational expressions involves dividing one rational expression by another, resulting in a new rational expression. Examples include 1 / (2x2) ÷ 1 / (3x) = 3 / 2x and 2 / (x + 1) ÷ (3 / (x - 1)) = 2(x - 1) / 3(x + 1). This operation is crucial for manipulating and simplifying rational expressions in algebra.

Topic

Rationals and Radicals

Description

An extraneous solution is a false solution discarded during the solving process. For example, solving √(x + 1) = x - 5 gives x = 3 or x = 8, but x = 3 is extraneous as it does not satisfy the original equation. This concept highlights the need to verify solutions in equations involving radicals or rational expressions.

Topic

Rationals and Radicals

Description

An extraneous solution is a false solution discarded during the solving process. For example, solving √(x + 1) = x - 5 gives x = 3 or x = 8, but x = 3 is extraneous as it does not satisfy the original equation. This concept highlights the need to verify solutions in equations involving radicals or rational expressions.

Topic

Rationals and Radicals

Description

Factoring a radical expression involves extracting whole number factors. For example, √(36x2) = 6x or solving x = (-6 ± √(36 - 16)) / 4, which simplifies to -3 ± √5 / 2. Factoring radicals is a key step in simplifying expressions and solving equations with square roots.

Topic

Rationals and Radicals

Description

Factoring a radical expression involves extracting whole number factors. For example, √36x2 = 6x or solving x = (-6 ± √(36 - 16)) / 4, which simplifies to -3 ± √5 / 2. Factoring radicals is a key step in simplifying expressions and solving equations with square roots.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

The fourth root of a number, when multiplied by itself four times, equals the given number. Examples include ∜16 = 2 and ∜81 = 3. Fourth roots extend the concept of radicals to higher powers, useful in algebraic and geometric contexts.

Understanding this video is essential for grasping the fundamentals of Rationals and Radicals. The mathematical concepts explored here form the basis for further exploration and application in related fields.

Topic

Rationals and Radicals

Description

The fourth root of a number, when multiplied by itself four times, equals the given number. Examples include ∜16 = 2 and ∜81 = 3. Fourth roots extend the concept of radicals to higher powers, useful in algebraic and geometric contexts.

This video includes Spanish audio and can be used with multilingual learners.

This is the transcript for the video of same title. Video contents: In this episode of Algebra Applications, students explore various scenarios that can be explained through the use of rational functions. Such disparate phenomena as submarines, photography, and the appearance of certain organisms can be explained through rational function models.

This is the transcript for the video of same title. Video contents: In spite of their massive size, submarines are precision instruments. A submarine must withstand large amounts of water pressure; otherwise, a serious breach can occur. Rational functions are used to study the relationship between water pressure and volume. Students graph rational functions to study the forces at work with a submarine.

This is the transcript for the video of same title. Video contents: All living things take up a certain amount of space, and therefore have volume. They also have a certain amount of surface area. The ratio of surface area to volume, which is a rational function, reveals important information about the organism. Students look at different graphs of these functions for different organisms.

This is the transcript for the video of same title. Video contents: The Hubble Telescope has transformed how we view the universe. We learn about the lens formula and how it is used in the construction of telescopes.

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This is the transcript for the video of same title. Video contents: After briefly reviewing the concept of inverse variation, this video explores Boyle??s law, a real world example of an inversely proportional relationship between pressure and volume of a gas. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, it goes on to examine similarities and differences among rational functions and numbers. Finally, it takes a look at rational functions graphs and ends with a delightful example merging Euclidean and analytic geometry, thanks to the TI-Nspire technology. Concepts explored: functions, rational expressions, rational functions, asymptotes

This is the transcript for the video of same title. Video contents: In this Investigation we look at an application of rational functions: Boyle's Law.

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This is the transcript for the video of same title. Video contents: In this Investigation we look at graphs of rational functions.

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What Are Rational Numbers?

This is the transcript that goes with the video segment entitled Video: Rational Numbers: Comparing and Ordering Rational Numbers.

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This is the transcript that goes with the video segment entitled Video: Rational Numbers: Dividing Rational Numbers.

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This is the transcript that goes with the video segment entitled Video: Rational Numbers: Multiplying Rational Numbers.

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This is the transcript that goes with the video segment entitled Video: Rational Numbers: Numerical Expressions with Rational Numbers.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Absolute Value.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Exponents.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Irrational Numbers.

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Distance Measures on a Number Line

This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers On the Cartesian Coordinate System.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Subtracting Rational Numbers.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Variable Expressions with Rational Numbers.

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What Are Rational Numbers?